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user 766
Questions that are about research in mathematics, or about the job of a research mathematician, without being mathematical problems or statements in the strictest sense. Do not use this tag for easy or supposedly easy mathematical questions.
57
votes
What mathematical problems can be attacked using DeepMind's recent mathematical breakthroughs?
This is a bit speculative, and perhaps too challenging for an undergraduate project, but I wonder if an AlphaGeometry type approach might be possible for the task of automatically upper bounding sums …
- 114k
26
votes
Oddities of evenness
The hairy ball theorem is only valid for even-dimensional spheres (or odd-dimensional ambient Euclidean spaces).
Similarly, the strong Huygens principle is only valid in odd-dimensional physical space …
347
votes
Accepted
What are the benefits of writing vector inner products as $\langle u, v\rangle$ as opposed t...
Mathematical notation in a given mathematical field $X$ is basically a correspondence
$$ \mathrm{Notation}: \{ \hbox{well-formed expressions}\} \to \{ \hbox{abstract objects in } X \}$$
between mathem …
22
votes
Examples of "unsuccessful" theories with afterlives
(Converted from a comment to an answer as requested.)
Non-Euclidean geometry was initially developed in hopes of deriving the parallel postulate from the other axioms of Euclidean geometry, as can be …
10
votes
Examples of problems where considering "discrete analogues" has provided insight or led to a...
In my paper
Tao, Terence, Norm convergence of multiple ergodic averages for commuting transformations, Ergodic Theory Dyn. Syst. 28, No. 2, 657-688 (2008). ZBL1181.37004.
I was able to settle a ques …
9
votes
What is the most useful non-existing object of your field?
Non-trivial approximate subrings of ${\bf R}$ or of ${\bf F}_p$.
The existence of such objects is ruled out by a number of "sum-product theorems", a typical one of which asserts that given a subset $ …
7
votes
Accepted
"Universal" differential identities
Let's work in two dimensions for notational simplicity. We claim that there is no non-trivial polynomial identity of the form
$$ P( f, f_x, f_y, f_{xx}, f_{xy}, \dots ) = 0$$
relating some finite num …
- 114k
24
votes
Accepted
Why is free probability a generalization of probability theory?
Quite a lot of questions here!
It is perhaps worth making a distinction between scalar classical probability theory - the study of scalar classical random variables - and more general classical proba …
- 114k
58
votes
Accepted
Why do people use "formal calculation" to describe informal calculations?
Formal, adj. Relating to or involving outward form or structure, often in contrast to content or meaning.
In mathematics, a formal argument is one that manipulates the form of an expression with …
13
votes
Accepted
Are reduced residue systems relative primorials an active area of research? If not, why not?
The Chinese remainder theorem tells us that the residue class ring ${\bf Z}/p\# {\bf Z}$ is isomorphic (as a ring) to the product of the finite fields ${\bf Z}/q {\bf Z}$, where $q$ ranges over the pr …
- 114k
252
votes
Examples of unexpected mathematical images
The third image below was certainly unexpected for my soon-to-be-collaborators, Emmanuel Candes and Justin Romberg. They started with a standard image in signal processing, the Logan-Shepp phantom:
…
62
votes
Analogues of P vs. NP in the history of mathematics
This isn't an exact analogue to P != NP, in which two large classes exist and it is undecided whether they are equal or not; instead, two large "universes" exist, of which only one is the truth, with …
23
votes
An example of a proof that is explanatory but not beautiful? (or vice versa)
I'd say that the proof of the four-colour theorem (particularly the "first generation" proof of Appel and Haken) is explanatory (we see that the source of four-colourability is the presence of unavoid …
19
votes
Trichotomies in mathematics
After passing to a subsequence if necessary, a sequence of real numbers either (a) converges to a real number; (b) diverges to $+\infty$; or (c) diverges to $-\infty$. In a similar vein, a sequence …
14
votes
How do you present a non-existence theorem?
One important class of "non-existence theorems", which includes Liouville's theorem as a model example, are the various rigidity theorems throughout mathematics that tend to have the general flavour o …